Lipschitz Regularity of the Eigenfunctions on Optimal Domains

We study the optimal sets (Formula presented.) for spectral functionals of the form (Formula presented.), which are bi-Lipschitz with respect to each of the eigenvalues (Formula presented.) of the Dirichlet Laplacian on Ω, a prototype being the problem min(Formula presented.) We prove the Lipschitz regularity of the eigenfunctions u1…up, of the Dirichlet Laplacian on the optimal set Ω*and, as a corollary, we deduce that Ω*is open. For functionals depending only on a generic subset of the spectrum, as for example λk(Ω), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.